The present invention relates to image segmentation. In particular it relates to efficient segmentation of images with smooth intensity regions.
The extraction of piecewise smooth regions from an image is of great interest in different domains, and still remains a challenging task. For example, this is very useful in medical imaging where organs or structures of interest are often characterized by smooth intensity regions. This problem has been formulated as the minimization of an energy by Mumford and Shah in: D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42:577-685, 1989 as:EMS(u, Γ)=μ2∫Ω(u0−u)2dx+∫Ω\Γ|∇u|2dx+ν|Γ|  (1)where u is the piecewise smooth function, Γ is the interface between smooth regions, and u0 is the original image. The interpretation of the three terms is straightforward: the first one is the usual mean-square data term; the second one means that one wants to extract smooth regions; the third one means that one want to extract regions with smooth boundaries.
The minimizer of this so-called Mumford-Shah functional gives a boundary that separates the image domain in smooth regions. A very interesting property of this approach is that it solves two common image-processing tasks simultaneously: image denoising and image segmentation. However, finding the minimizer is not straightforward and remains an issue. Being non-convex, the functional is most of the time minimized using gradient descent techniques which are subject to local minima. For example, in “L. Vese and T. Chan. A multiphase level set framework for image segmentation using the mumford and shah model. International Journal of Computer Vision, 50:271-293, 2002” and “A. Tsai, A. Jr. Yezzi, and A. S. Willsky. Curve evolution implementation of the mumford-shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Transactions on Image Processing, 10(8):1169-1186, August 2001”, the optimization process alternates between the evolution of one or two level set functions such as provided in “S. Osher and J. Sethian. Fronts propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. of Comp. Phys., 79:12-49, 1988”, “J. Sethian. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Sciences. Cambridge Monograph on Applied and Computational Mathematics. Cambridge University Press, 1999” and “A. Dervieux and F. Thomasset. A finite element method for the simulation of Raleigh-Taylor instability. Springer Lect. Notes in Math., 771:145-158, 1979” and the resolution of Poison partial differential equations. This process is computational extensive and requires a very good initialization to avoid being stuck into local minima.
To relax this problem, one can consider a restriction of EMS to piecewise constant functions. Let Ωi be the open subsets delimited by Γ, the piecewise constant Mumford-Shah functional writes:
                                          E            0            MS                    ⁡                      (            Γ            )                          =                                            ∑              i                        ⁢                                          ∫                                  Ω                  i                                            ⁢                                                                    (                                                                  u                        0                                            -                                                                        mean                                                      Ω                            i                                                                          ⁡                                                  (                                                      u                            0                                                    )                                                                                      )                                    2                                ⁢                                  ⅆ                  x                                                              +                      v            ⁢                                        Γ                                                                        (        2        )            This functional was shown in “D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42:577-685, 1989” to be a limit functional of expression (1) as μ→0. A level set implementation of this functional known as the Chan and Vese model was proposed in “L. Vese and T. Chan. A multiphase level set framework for image segmentation using the mumford and shah model. International Journal of Computer Vision, 50:271-293, 2002”. While this simplified functional is easier to minimize, it also makes a very strong assumption on the image by assuming implicitly a Gaussian intensity distribution for each region Ωi as described in “M. Rousson and R. Deriche. A variational framework for active and adaptative segmentation of vector valued images. In Proc. IEEE Workshop on Motion and Video Computing, pages 56-62, Orlando, Fla., December 2002”. Other papers model this distribution with Gaussian mixtures such as in “N. Paragios and R. Deriche. Geodesic active regions and level set methods for supervised texture segmentation. The International Journal of Computer Vision, 46(3):223-247, 2002” and “0. Juan, R. Keriven, and G. Postelnicu. Stochastic motion and the level set method in computer vision: Stochastic active contours. The International Journal of Computer Vision, 69(1):7-25, 2006”, or with nonparametric distributions such as “J. Kim, J. Fisher, A. Yezzi, M. Cetin, and A. Willsky. Nonparametric methods for image segmentation using information theory and curve evolution, in IEEE International Conference on Image Processing, pages 797-800, September 2002”, but they all make the assumption of a global distribution over each region. In many real images, these global intensity models are not valid. This is often the case in medical imaging, especially in MR images where an intensity bias can be observed.
Several approaches are available to overcome the limitation of global techniques. One is to consider image gradients by fitting the contour to image discontinuities. This is generally referred to as edge-based methods, and it is the basis of the Geodesic Active Contours as described in “V. Caselles, R. Kimmel, and G. Sapiro. Geodesic active contours. International Journal of Computer Vision, Vol 22:61-79, 1997” and “S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi. Gradient flows and geometric active contour models. In Proceedings of the 5th International Conference on Computer Vision, pages 810-815, Boston, Mass., June 1995. IEEE Computer Society Press”. A third functional was also introduced in the seminal work of Mumford and Shah: “D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42: 577-685, 1989”. This functional is the integral along I′ of a generalized Finsler metric and leads indeed to the first geodesic active contour (before “V. Caselles, R. Kimmel, and G. Sapiro. Geodesic active contours. International Journal of Computer Vision, Vol 22:61-79, 1997” and “S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi. Gradient flows and geometric active contour models. In Proceedings of the 5th International Conference on Computer Vision, pages 810-815, Boston, Mass., June 1995. IEEE Computer Society Press”.).
Edge-based methods are also well-known for their high sensitivity to noise and for the presence of local minima in the optimization as described in D. Cremers, M. Rousson, and R. Deriche. A review of statistical approaches to level set segmentation: integrating color, texture, motion and shape. International Journal of Computer Vision, 2006. Another alternative was briefly discussed in “L. Vese. Multiphase object detection and image segmentation, in S. Osher and N. Paragios, editors, Geometric Level Set Methods in Imaging, Vision and Graphics, pages 175-194. Springer Verlag, 2003” where the function u0 of the Mumford-Shah functional is restricted to a linear function of the spatial location x: u0(x)=a.x+b. Even though this last one is promising, it is still restricted to very particular spatial distributions of the intensity.
Accordingly methods and systems for extracting piecewise smooth regions but with improved performance are required.